Household Intertemporal Behavior: a Collective Characterization and Empirical Tests

Maurizio Mazzocco ∗ University of Wisconsin-Madison

∗I am very grateful to Orazio Attanasio, Pierre Andre Chiappori, James Heckman, John Kennan, Lars Lefgren, Annamaria Lusardi, Bernard Salanie and Robert Townsend for their insight and suggestions. I would also like to thank the participants at the 27th Seminar of the European Group of Risk and Insurance Economists, at the 2002 AEA Meeting, at the Southeast Economic Theory and International Economics Meetings and at Seminars at the University of Chicago, University College of London, University of California at Irvine, Ohio State University, University of Wisconsin-Madison, Western Ontario University, for helpful comments. Errors are mine.

1 Abstract

In this paper, each household is characterized as a group of agents making joint decisions. Under the assumption of full efficiency, it is shown that households with several adults can be characterized by a unique utility function if and only if individual members have identical discount factors, HARA utility functions with identical curvature parameter and identical beliefs. If this conditions are violated, the household objective function and consequently Euler equations depend on the decision power of each spouse. If the assumption of full efficiency is relaxed, I find that standard Euler equations of couples are replaced by inequalities, even if the restrictions are satisfied. However, the Euler equations of singles should be fulfilled with and without efficiency. Using the PSID as well as the CEX, these theoretical implications are tested. I find that the standard model of household intertemporal behavior is consistent with the consumption pattern of singles, but not with the consumption behavior of couples. Finally it is shown that full efficiency has different predictions for household Euler equations from the limited commitment version of the model. These predictions will be tested using both the PSID and the CEX.

2 1 Introduction

In this paper, it is shown that the standard model of household intertemporal behavior is consistent with the consumption pattern of US households with only one adult. However, the standard framework is rejected for US couples. These results and the approach employed in the paper are new in several respects. First, it is explicitly taken into consideration that households are composed by several individual members with different preferences. In most of the consumption literature, each household is characterized as if a single agent were making the decisions. Specifically, a unique utility function is maximized under a budget constraint. Using the alternative approach, it is shown that the testable implications of the standard intertemporal maximization model should be satisfied by households composed by only one adult, but rejected for household with several adults. Second, I consider a full commitment as well as a limited commitment version of intertemporal behavior and find that the two approaches have different implications for household Euler equations. Third, the theoretical restrictions of the intertemporal optimization models are tested using the Panel Study of Income Dynamics (PSID) as well as the Consumer Expenditure Survey (CEX). I find that the assignment of individual preferences to household members and a proper treatment of their interactions can be important theoretically as well as empirically. The PSID and CEX are the most frequently used data sets to test the theoretical restrictions of intertemporal optimization. In both data sets, the majority of the households are married or cohabiting couples. Consequently, the outcome of any test is driven by the pattern of consumption of households with several adults. Therefore the theoretical and empirical results of this paper represent a possible structural explanation of the rejections of the theory reported in the past 20 years. Hall and Mishkin (1982), for instance, find that food consumption is excessively sensitive to lagged labor income. Hayashi (1985) finds excess sensitivity using seven different consumption goods. Flavin (1981) and Campbell and Mankiw (1989) strongly reject the Euler equations using aggregate time-series data. In this paper, the intertemporal allocation of resources is modelled as the joint decision of household members. To that end, I characterize the household as a group of agents, each of them

3 being represented by individual preferences. By means of this framework, the following results are shown. a) Under the assumption of full efficiency, households with several adults can be characterized by a unique utility function if and only if strong restrictions on preferences are satisfied. Specifically, if and only if all household members have identical discount factors, Harmonic Ab- solute Risk Aversion utility functions with identical curvature parameter and identical beliefs. For instance, under the standard assumption of Constant Relative Risk Aversion utilities, these restrictions are satisfied if and only if individual members have identical risk aversion, hypothesis that is strongly rejected by Barsky, Juster, Kimball and Shapiro (1997). In all other cases, the only objective function that can be assigned to the household depends on the decision power of individual members and on all variables having an effect on it. Consequently, Euler equations of couples should depend on all the factors affecting the intra-household allocation of resources, such as individual income, differences in age and family composition. b) If the assumption of full efficiency is relaxed to allow for limited commitment, the results are even stronger. Euler equations of couples are replaced by supermartingales, even if the restrictions on preferences are satisfied. Therefore, tests of intertemporal optimization based on Euler equations should reject the classical model even if liquidity constraints are not binding. c) The full efficiency version of the model proposed in this paper has different implications for household Euler equations from the limited commitment version. Under full commitment, the factors affecting individual decision power should enter household Euler equations only as interaction terms with household consumption. If the assumption of full commitment is not satisfied, those factors should enter household Euler equations both directly and as interaction terms with household consumption. d) The presence of those factors in household Euler equations generates additional testable implications: the corresponding coefficients must satisfy proportionality conditions that are very unlikely to be fulfilled unless the model at stake is correct. e) Using the PSID and the CEX, I find that consumption of couples is excessively sensitive to household income. However, I do not find excess sensitivity for households with only one

4 adult. Therefore I can reject the model that assigns a unique utility function to the household independently of the family structure. But I cannot reject the alternative specification introduced in this paper. The empirical results are consistent with the findings of Attanasio and Weber (1995). In their insightful paper, Attanasio and Weber (1995) show that the intertemporal model of optimizing behavior for consumption is consistent with US micro data if household preferences are modelled so as to take into consideration changes in family composition and labor supply behavior over the life cycle. The model of household consumption introduced in this paper extends the collective model developed by Chiappori (1988, 1992) in two directions. The collective model is first generalized to a multi-period framework to analyze household intertemporal optimization. The extension to a multi-period framework raises a commitment issue. In the paper I consider two possible approaches to household commitment. In the first model, household members can commit in their first period to an allocation of resources for the future. In the second framework household members can only commit to an allocation of resources for the current period but not for the future. This distinction allows me to determine the impact of limited efficiency on Euler equations. Lund- berg, Startz and Stillman (forthcoming) use a three-period collective model with no uncertainty to explore the retirement-consumption puzzle. The topic analyzed in this paper is very different. The emphasis is on the relation between household intertemporal behavior and family structure. Moreover, the model that I propose is more general in several dimensions. Uncertainty is a crucial component of household behavior as emphasized by Mazzocco (2001) and modelled in this paper. I take into consideration the additive form of altruism. Finally, I allow for an arbitrary number of periods. Lundberg and Pollack (2001) use a non-stationary multi-stage game to analyze theoretically the location decision of a married couple. They show that marital decisions involving the future are in general not efficient. These results are consistent with the finding of this paper that Euler equations are replaced by inequalities when the assumption of full efficiency is relaxed. The approach of splitting the sample in two groups based on theoretical ground has been exten- sively used in the literature. For example, Bernanke (1984), Hayashi (1985), Zeldes (1989) and more recently Browning and Chiappori (1998) use this strategy. This paper differs from the pre-

5 vious literature as it allows for self selection. Moreover, this is the first paper that estimates Euler equations separately for singles and couples. The paper is organized as follows. In section 2, the full efficiency Intertemporal Collective Model (ICM) is introduced. Afterwards, the assumption of full efficiency is relaxed and the case of limited commitment is analyzed. It is shown that the full efficiency collective model is equivalent to a two-stage framework in which first the household allocates optimally total resources across individual members. Second, given the allocation of resources, each spouse maximizes her own utility subject to individual budget constraints. This alternative formulation is much more tractable theoretically as well as empirically. Section 3 establishes the conditions on preferences and efficiency under which the standard Euler equations are satisfied. Moreover, it discusses the consequences for the estimation of Euler equations if these restrictions are violated. In section 4, tests to evaluate the standard unitary model against the ICM are outlined. Section 5 describes the data. Section 6 discusses some econometric issues. Section 7 reports the estimation results. Some concluding remarks are presented in the final section.

2 The Intertemporal Collective Model

To answer several policy questions, precise estimates of the intertemporal and intratemporal rate of substitutions are required. In the past twenty years, economists have estimated these coefﬁ- cients using household Euler equations. With rare exceptions, all these works are based on the crucial assumption that household intertemporal behavior can be characterized as if a single agent were making the decisions. In reality, the majority of households are composed by several agents making joint decisions. In the CEX, about 66 percent of the interviewed households are married or cohabiting couples. A similar pattern is displayed in the PSID. It is therefore important to determine the impact of this assumption on the estimation of Euler equations. I will consider a household living for T periods and composed by n agents. Agent i is characterized by individual preferences over a private composite good, ci, and a public composite good,

Q, an individual discount factor, βi, and individual beliefs. Individual preferences are assumed

6 to be separable over time and to be represented by a von Neumann-Morgenstern utility function, ui ci,Qi. A probability measure F i (ω) describes the individual beliefs, where ω ∈ Ω is a po- tential state of the world. In each period t ∈ {0, ..., T } and state of the economy ω ∈ Ω, member i i is endowed with an exogenous stochastic income stream y (t, ω) t∈T,ω∈Ω. For any given (t, ω), the household can either consume or save. One risk-free asset is traded in the economy. s (t, ω) and R (t) denote respectively the amount of wealth invested in the risk-free asset and the gross return on the risk-free asset. Total household income and total household consumption are de- Pn i Pn i noted with Y (t, ω) = i=1 y (t, ω), C (t, ω) = i=1 c (t, ω). Let Z = {z1, ..., zm}, be the set of variables affecting the individual decision power. Throughout the paper, utility functions are assumed to be continuous, continuously differentiable, have positive ﬁrst derivative and negative second derivative. All functions are assumed to be measurable with respect to the probability measure F i.

2.1 The Unitary Model

Consumption is generally measured only at the household level. To overcome this obstacle, most of the literature on intertemporal optimization assumes that household preferences can be represented by a unique von Neumann-Morgenstern utility function, U (C,Q). Hence, the household allocates resources over time and across states of the world according to the following program:

" T # X t max E β U (Ct,Qt) (1) {C ,Q ,s } t t t t∈T,ω∈Ω t=0 n X i s.t. Ct + PtQt + st ≤ yt + Rtst−1 ∀t, ω i=1

sT ≥ 0 ∀ω.

This approach overlooks that the household is composed by several agents with different preferences, interacting in a complex way. In particular, in this framework the distribution of decision power between members is not explicitly modelled and only full efﬁciency can be considered.1

1Becker (1991) offers a justiﬁcation for this approach.

7 2.2 The Full-Efﬁciency Intertemporal Collective Model

To take into consideration that households are composed by several agents making joint decisions, each individual member is represented by individual preferences. Building on the seminal work of Chiappori (1988, 1992), it is assumed that household decisions are efﬁcient, i.e. the corresponding allocation of resources is always on the Pareto frontier. In an intertemporal framework, household efﬁciency requires two assumption. First, members must be able to commit at t = 0 to an allocation of resources for any future time, t, and state of nature, ω. Second, household members must have symmetric information. Under these assumptions, household intertemporal decisions can be represented by means of the following Pareto problem:

" T # 1 X t 1 1 max E β u (c ,Qt) (2) i=1,...,n 1 t ci,Q ,s { t t t}t∈T,ω∈Ω t=0

" T # i X t i i s.t. µi : E βi u (ct,Qt) ≥ ui (Z) i = 2, ..., n t=0 n n X i X i c + PtQt + st ≤ yt + Rtst−1 ∀t, ω i=1 i=1

sT ≥ 0 ∀ω,

for some set of reservation utilities {ui}i=2,...,n. In general, each ui is a function of all the variables affecting the individual decision power, Z. For a given Z, the set of efﬁcient outcomes obtains as

{ui}i=2,...,n varies within its domain. The full commitment assumption is implicit in the participation constraints of problem (2). Indeed, the household commits to an allocation at t = 0 and sticks to it for the next T periods even if the relative power of the n members changes over time.2 The full efﬁciency ICM can be formulated using an alternative two-stage framework. In the ﬁrst stage, the household decides how much to consume of the public good and the optimal distribution of total income between the n agents for each pair (t, ω). In the second stage, each

2It is useful to note that with this framework it is possible to take into consideration the additive form of altruism,

i 1 i n i i X j j U c , .., c , .., c ,Q = u (c ,Q) + δjiu (c ,Q). j6=i

8 agent chooses individual private consumption and savings, given her share of total income and the chosen quantity of the public good for each (t, ω). The two-stage approach is appealing for two reasons. It is mathematically and empirically more tractable. Moreover, the ﬁrst stage gives a complete description of the efﬁcient distribution of available resources between the n members. i i=1,...,n Formally, let ρ (t, ω) t∈T,ω∈Ω be an arbitrary distribution of available non labor income among the n members for each pair (t, ω). Let {Q (t, ω)}t∈T,ω∈Ω be an arbitrary consumption of the i i=1,...,n public good for each pair (t, ω). Then, given ρ (t, ω) t∈T,ω∈Ω and {Q (t, ω)}t∈T,ω∈Ω, in the second stage agent i solves the following program:

" T # X V i(ρi ,Q ) = max Ei βtui(ci,Q ) t,ω t,ω t∈T,ω∈Ω i t t (4) ci,si { t t}t∈T,ω∈Ω t=0

i i i i s.t. ct + st ≤ ρt + Rtst−1 ∀ (t, ω)

sT ≥ 0 ∀ω.

In the ﬁrst stage, the household determines the optimal allocation of total income between the n members, solving the following program:

To see this note that (2) can be written using the Kuhn-Tucker multipliers associated with the participation constraints:

" T # " T # 1 X t 1 1 n X t n n max µ¯ E0 β1u (ct ,Q) + .. +µ ¯ E0 βnu (ct ,Q) (3) ci,Q,s { }i=1,...,n t=0 t=0 n n X i X i ct + PtQt + st ≤ yt + Rtst−1 ∀ (t, ω) i=1 i=1

sT ≥ 0 ∀ω, ∈ Ω

This formulation is equivalent to a model with additive altruism in which µ¯1, ..., µ¯n are the sum of Pareto weights and altruism parameters.

9 1 1 max V ( ρ ,Qt,ω ) (5) i=1,...,n t,ω t∈T,ω∈Ω ρi,Q { t t}t∈T,ω∈Ω

s.t. V i(ρi ,Q ) ≥ u i = 2, ..., n t,ω t,ω t∈T,ω∈Ω i n n X i X i ρt + PtQt ≤ yt ∀ (t, ω) . i=1 i=1 The following proposition states that the two-stage interpretation is equivalent to the full efﬁ- ciency intertemporal collective model.

ci ,Q , s i=1,...,n s = Proposition 1 Let t,ω t,ω t,ω t∈T,ω∈Ω be a solution of the two-stage model. Let t,ω Pn si (t, ω) ci ,Q , s , i=1,...,n i=1 t,ω for every . Then t,ω t,ω t,ω t∈T,ω∈Ω is a solution of the ICM, (2). Let ci ,Q , s i=1,...,n ρi i=1,...,n t,ω t,ω t,ω t∈T,ω∈Ω be a solution of the ICM, (2). Then there exist a sharing rule t,ω t∈T,ω∈Ω, {Q } si i=1,...,n s = consumption of the public good t,ω t∈T,ω∈Ω and individual savings t,ω t∈T,ω∈Ω, with t,ω Pn si (t, ω) ρi , ci , si i=1,...,n , i=1 t,ω for every , such that t,ω t,ω t,ω t∈T,ω∈Ω is a solution of the two-stage model.

Proof. First and Second Welfare Theorem with production.

2.3 The Limited-Commitment ICM

In the full efﬁciency ICM, it is assumed that individual members have symmetric information and can commit to an allocation of resources for the future. The assumption of symmetric information is likely to be satisﬁed for most households. The assumption on commitment is much stronger. To verify the effect of limited commitment on household intertemporal behavior, in this section I will consider a framework in which household members can commit within periods but not on future plans, i.e. the intertemporal allocation of resources must satisfy individual participation constraints for each period and state of the world. Formally, the household behaves according to

10 the following program:

" T # " T # 1 X t 1 1 2 X t n n max E0 β1µ1u (ct ,Qt)+ ... + E0 βnµnu (ct ,Qt) (6) ci,Q ,s { t t t}t∈T,ω∈Ω t=0 t=0 " # i XT −τ t i i i s.t. µi (τ, ω): Eτ βi u (ct+τ ,Qt+τ ) ≥ ui,τ,ω (sτ−1) ∀ω, τ > 0, i = 1, ..., n t=0t=0 T T X i X i ct + PtQt + st ≤ yt + Rtst−1 ∀ (t, ω) i=1 i=1

bT + sT ≥ 0 ∀ω.

u (s ) i=1,...,n for some set of reservation utilities i,t,ω t−1 t∈T,ω∈Ω, which may depend on household savings and more generally on Z. The main difference between the full efﬁciency and limited commitment ICM is summarized i=1,...,n by the multipliers {µi}i=1,..n in (2) and the set of multipliers {µi (t, ω)}t∈T,ω∈Ω in (3). µi can be interpreted as the decision power of member i at t = 0 in the collective decision process. For any pair (t, ω), µi (t, ω) is the decision power of member i at (t, ω). Thus the solution of (2) is only affected by the distribution of decision power between the n members at t = 0. Whereas the optimal allocation in (3) is a function of the distribution of power between the n members at each pair (s, ω).

3 Household Euler Equations

In the past twenty years, Euler equations have been derived and estimated using the unitary model. Speciﬁcally, household intertemporal behavior has been tested by means of the following equation:

UC (Ct,Qt) = βEt [UC (Ct+1,Qt+1) Rt+1] . (7)

Household Euler equations can also be derived using the ICM. Under the assumption of full ef- ﬁciency and separability over time and across states of nature, it is always possible to construct

11 household preferences solving the following representative agent problem:

1 1 1 n n n V (C, Q, µ(Z)) = max β1F (ω)µ1 (Z) u c ,Q + ... + βnF (ω)µn (Z) u (c ,Q) i {c ,Q}i=1,...,n n X s.t. ci + PQ = C i=1 where µ(Z) = {µi (Z)}i=1,...,n are the individual Pareto weights. It is then straightforward to derive the following household Euler equation using standard arguments:

Vc (Ct,Qt, µ(Z)) = βEt [Vc (Ct+1,Qt, µ(Z)) Rt+1] . (8)

If the assumption of full commitment is not satisﬁed, it is still possible to derive household preferences and the corresponding Euler equations using the approach developed in Marcet and Mari- mon (1992) and Marcet and Marimon (1998). It is possible to show that the limited commitment ICM can be rewritten using the following formulation:

T T X 1 t 1 1 X n t n n max E0 β1M1,tu (ct ,Qt) − µ1,nu1,t + ... + E0 βnMn,tu (ct ,Qt) − µn,tun,t i {ct,Qt,st} t=0 t=0 T T X i X i ct + PtQt + st ≤ yt + Rtst−1 ∀ (t, ω) i=1 i=1

sT ≥ 0 ∀ω,

Pt where Mi,t = τ=0 µi,t, µi,t is the Kuhn-Tucker multiplier corresponding to the participation constraint of member i at time t and µi,0 is the initial Pareto weight of agent i. Under the assumption that ui,t is independent of savings and other decision variables, it is possible to exploit the separability of preferences over time and across states of the world to determine household preferences by solving a representative agent problem at each t:

1 1 1 n n n V (C, Q, Mt(Z)) = max β1F (ω)M1,t (Z) u c ,Q + ... + βnF (ω)Mn,t (Z) u (c ,Q) i {c ,Q}i=1,...,n n X s.t. ci + PQ = C. i=1

12 Consequently, if the assumption of limited commitment is not satisﬁed, household Euler equations can be written in the form,

Vc (Ct,Qt,Mt(Z)) = βEt [Vc (Ct+1,Qt,Mt(Z)) Rt+1] , (9) i.e. Euler equations still depend on individual decision power, but the distribution of decision power can change over time if the individual outside options vary. Given the extensive use of the unitary approach to estimate Euler equations it is important to determine under which restrictions individual preferences can be aggregated in a unique utility function so that,

Vc (Ct,Qt, µ(Z)) = UC (Ct,Qt) (10) or alternatively,

Vc (Ct,Qt,Mt(Z)) = UC (Ct,Qt) (11)

Moreover, if these conditions are fulﬁlled, it is crucial to determine which additional assumptions are required for household Euler equations to be satisﬁed.

i i i i Deﬁnition 1 For any given level of Q, deﬁne uQ c = u c ,Q . An ISHARA household is a household satisfying the following conditions: (i) all members have identical discount factor β and beliefs F (ω); (ii) for i = 1, ..., n,

i −γi i0 i c uQ c = ai + γi with γ1 = ... = γn, i.e. individual preferences conditional on the public good are Harmonic Absolute Risk Aversion with identical curvature parameter.3

It is worth noting that the assumption of HARA preferences with identical shape parameter is very restrictive. For instance, if all agents have CRRA preferences, the household is ISHARA

3 i i Note that in the conditional utility function uQ c , ai and γi will be in general complex functions of Q. Therefore preferences will not be in general separable between Q and ci even under the assumption that the household belongs to the ISHARA class.

13 if and only if all members have identical preferences. The following result is a generalization of Gorman aggregation to an intertemporal framework and states that an ISHARA household is a sufﬁcient and necessary condition for a unique utility function to exist.

Assumption 1 Each agent has strictly positive Pareto weights, i.e. {µi}i=1,...,n.

Assumption 2 For each i = 1, ..., n, limui0 (x) = ∞. x→0

Assumption 3 For each i = 1, ..., n, the probability measure Fi (ω) has a density fi (ω).

Theorem 1 The household can be represented using a unique utility function that does not depend on the Pareto weights if and only if the household belongs to the ISHARA class.

Proof. In the appendix. To understand theorems 1, observe that the unitary and collective frameworks are equivalent if and only if the optimal allocation of total income between members has no effect on the aggregate behavior of the household. Intuitively, for this to happen two conditions must be satisfied. First, each agent’s income expansion path must be linear. If not, exploiting the nonlinearity, it is always possible to find two households with identical aggregate incomes, but different allocation of resources and hence different intertemporal behavior. This has an important consequence: even if all agents have identical utility functions, the standard and collective framework are not necessarily equivalent. Second, the slope of the linear income expansion path must be identical across agents. The strong implication is that household members must have the same sensitivity to risk and uncertainty and this is not supported by empirical evidence. For instance, Barsky, Juster, Kimball and Shapiro (1997) find that the correlation of relative risk tolerance across household members is only 0.12. Only ISHARA households satisfy both conditions. The following result is a simple corollary of theorem 1 and it claims that under the assumptions of full efficiency and ISHARA households the standard Euler equation (7) is satisfied.

i i=1,...,n Corollary 1 Let c (t, ω) ,Q (t, ω) t∈T,ω∈Ω be the solution of the full efﬁciency ICM. Set C (t, ω) = n Pci (t, ω) for every (t, ω). Assume that the household belongs to the ISHARA class. Then, the i=1

14 following household Euler equation is satisﬁed:

UC (C (t, ω) ,Q (t, ω)) = βEt [UC (C (t + 1, ω) ,Q (t + 1, ω)) Rt+1] .

Proof. In the appendix. This result suggest that, for non-ISHARA households, the only aggregator that can be used to characterize household preferences will depend on the distribution of decision power and on the set of variables having an effect on it. It is important to establish whether this result still applies if full efﬁciency is replaced by limited commitment. In particular, suppose that households belong to the ISHARA class, but individual members cannot commit for the future. The following result states that household Euler equations are replaced by supermartingales, even if the conditions for Gorman aggregations are satisﬁed.

Assumption 4 The reservation utility of each member, in each period and state of the world is and increasing function of household savings, i.e. ∂u (ω) i,t+1 ≥ 0 for any i, t, ω. (12) ∂st (ω)

Theorem 2 Consumption Euler equations are replace by consumption supermartingales,4

UC (C (t, ω) ,Q (t, ω)) > βEt [UC (C (t + 1, ω) ,Q (t + 1, ω)) Rt+1] , (13)

Two remarks are in order. First of all, condition (13) resembles in part the results obtained in the literature on limited commitment.5. In this literature, the inequality applies to each agent, but not necessarily to the group. Theorem 2 suggests that a similar inequality applies to the household, if it belongs to the ISHARA class. Second, condition (13) is isomorphic to the ﬁndings of the literature on liquidity constraints. Consequently, a test designed to detect liquidity constraints using this inequality has no power against the alternative of limited commitment.

4Intuitively savings should have a positive effect on individual outside options. If savings affect negatively the reservation utility of at least one agent, then consumption Euler equations are still replaced by inequalities, but the direction of the inequality depends on the relative effect of savings on individual outside options 5See for example Ligon, Thomas and Worral (1990), Kocherlakota (1996) and Attanasio and Rios-Rull (2000)

15 In the next section, these results will be used to set up three tests to verify the empirical relevance of the ICM.

4 Testing Household Intertemporal Behavior

The unitary model and the ICM generate different household Euler equations. Since household Euler equations are generally highly non-linear it is difficult to determine how these differences affect household Euler equations. A possible solution is to linearize the Euler equations using a Taylor expansion. In this section, I will restrict the analysis to households with 2 decision makers. At the same time, I will generalize the analyzes to consider also household Euler equations for the public good. In the following theorem, log-linearized household Euler equations are derived using the full efficiency ICM. Let φ1 and φ2 be defined as follows: n n o n o n oo φ1 C,ˆ Q,ˆ Zˆ = ln VC exp ln CEˆ [C] , exp ln QEˆ [Q] , κ exp ln ZEˆ [Z] n n o n o n oo φ2 C,ˆ Q,ˆ Zˆ = ln VQ exp ln CEˆ [C] , exp ln QEˆ [Q] , κ exp ln ZEˆ [Z]

ˆ C ˆ Q ˆ Z where C = ln E[C] , Q = ln E[Q] , Z = ln E[Z] and κ is equal to µ if the full efﬁciency ICM is considered and equal to M if the limited commitment ICM is considered.

ˆ ˆ 3 2 m Assumption 5 φi C, Q, Z :Θi → R, where φi ∈ C , Θi ∈ R × R is open, convex and

0 ∈ Θi, for i=1,2.

Theorem 3 Under assumption 5, the household Euler equations for the full efﬁciency ICM can be written as follows:

m m Ct+1 Qt+1 X zi Ci X zi Qt ln = α + α ln R + α ln + α ln ln + α ln ln C 0 1 t+1 2 Q i,3 zˆ ˆ i,4 zˆ ˆ t t i=1 i Ci i=1 i Q " 2 2# " 2 2# Ct+1 Ct Qt+1 Qt +α5 ln − ln + α6 ln − ln Cˆ Cˆ Qˆ Qˆ Ct+1 Qt+1 Ct Qt +α7 ln ln − ln ln + RC C,ˆ Q,ˆ Zˆ + ln (1 + et+1,C ), Cˆ Qˆ Cˆ Qˆ

16 m m Qt+1 Ct+1 X zi Ct+1 X zi Qt+1 ln = δ + δ ln R + δ ln + δ ln ln + δ ln ln Q 0 1 t+1 2 C i,3 zˆ ˆ i,4 zˆ Q t t i=1 i Ct i=1 i t " 2 2# " 2 2# Ct+1 Ct Qt+1 Qt +δ5 ln − ln + δ6 ln − ln Cˆ Cˆ Qˆ Qˆ Ct+1 Qt+1 Ct Qt +δ7 ln ln − ln ln + RQ C,ˆ Q,ˆ Zˆ + ln (1 + et+1,Q), Cˆ Qˆ Cˆ Qˆ with ¯ ¯ VC VCCµ − VCC VCµ ∂µ Z VC VCQµ − VCQVCµ ∂µ Z αi,3 = C¯z¯i, αi,4 = Q¯z¯i, VCC VC ∂zi VCC VC ∂zi ¯ ¯ VQVQCµ − VQC VQµ ∂µ Z VQVQQµ − VQQVQµ ∂µ Z δi,3 = C¯z¯i, δi,4 = Q¯z¯i, VQQVQ ∂zi VQQVQ ∂zi where C¯ = E[C], Q¯ = E[Q], z¯i = E[zi], RC and RQ are the remainders of the Taylor’s formula and eC and eQ are the expectation errors.

Proof. In the appendix. It is interesting to note that using the full efficiency ICM, the factors determining individual decision power enter the household Euler equations only as interaction terms with the private and public good. To have some insight note that under the assumption of separability over time and across states of nature, the household intertemporal optimization can be analyzed by considering a specific period and state of nature at a time. Under the assumption of full efficiency only the relative power of household members at time t = 0 is relevant in the decision process, i.e. any change in individual outside options has no effect on the allocation of resources. Hence, each period and state of nature is characterized by an identical function µ(Z). Consequently, a change 0 0 00 00 in one of the factors, zi, has a different effect on (t , ω ) relative to (t , ω ) only because optimal consumption at (t0, ω0) differs from the optimal consumption at (t00, ω00), i.e. for each (t, ω) the change in µ(Z) is identical but it occurs at a different point of the (t, ω)-Pareto frontier. Since the Taylor expansion is calculate at the average value of C, Q and Z for each pair (t, ω), the first order term in zi will be equal to zero. All the effect is captured by the interaction terms, which evaluate how the average value of optimal consumption differ from the actual value.

17 To be able to distinguish between the full efﬁciency ICM and the limited commitment ICM, it is important to derive a log-linearized version of the household Euler equation with limited commitment. The following theorem establishes that the factors affecting the individual decision power enter the intertemporal optimization condition, not only as interaction terms with private and public consumption as in the full efﬁciency case, but also directly.

Theorem 4 Under assumption 5, the household Euler equation for the limited commitment ICM can be written as follows:

m m m Ct+1 Qt+1 X zi X zi Ci X zi Qt ln = α + α ln R + α ln + α ln + α ln ln + α ln ln C 0 1 t+1 2 Q 3 zˆ i,4 zˆ ˆ i,5 zˆ ˆ t t i=1 i i=1 i Ci i=1 i Q m 2 " 2 2# " 2 2# X zi Ct+1 Ct Qt+1 Qt + α ln + α ln − ln + α ln − ln 6 zˆ 7 ˆ ˆ 8 ˆ ˆ i=1 i C C Q Q Ct+1 Qt+1 Ct Qt +α9 ln ln − ln ln + RC C,ˆ Q,ˆ Zˆ + ln (1 + et+1,C ), Cˆ Qˆ Cˆ Qˆ

m m m Qt+1 Ct+1 X zi X zi Ct+1 X zi Qt+1 ln = δ + δ ln R + δ ln + δ ln + δ ln ln + δ ln ln Q 0 1 t+1 2 C 3 zˆ i,4 zˆ ˆ i,5 zˆ Q t t i=1 i i=1 i Ct i=1 i t m 2 " 2 2# " 2 2# X zi Ct+1 Ct Qt+1 Qt + δ ln + δ ln − ln + δ ln − ln 6 zˆ 7 ˆ ˆ 8 ˆ ˆ i=1 i C C Q Q Ct+1 Qt+1 Ct Qt +δ9 ln ln − ln ln + RQ C,ˆ Q,ˆ Zˆ + ln (1 + et+1,Q), Cˆ Qˆ Cˆ Qˆ with ¯ ¯ VCM ∂µt Z VC VCCµ − VCC VCµ ∂µ Z αi,3 = z¯i, αi,4 = C¯z¯i, VCC ∂zi VCC VC ∂zi ¯ ¯ VC VCQµ − VCQVCµ ∂µ Z VQM ∂µ Z αi,5 = Q¯z¯i, δi,3 = z¯i, VCC VC ∂zi VQQ ∂zi ¯ ¯ VQVQCµ − VQC VQµ ∂µ Z VQVQQµ − VQQVQµ ∂µ Z δi,4 = C¯z¯i, δi,5 = Q¯z¯i, VQQVQ ∂zi VQQVQ ∂zi where C¯ = E[C], Q¯ = E[Q], z¯i = E[zi], RC and RQ are the remainders of the Taylor’s formula, Pt eC and eQ are the expectation errors, Mt = τ=0 µτ and µt is the Kuhn-Tucker multiplier at time t.

18 Proof. In the appendix. To understand why the factors Z enter directly the Euler equation in the limited commitment ICM, note that in this version of the model, the relative power of each household member can vary over time through changes in the individual outside options. Consequently, a variation in the factor zi will affect differently the distribution of decision power at time t, Mt, relative to the distribution of decision power at t + 1, Mt+1 = Mt + µt+1. The following result is a simple corollary of theorems 3 and 4.

Corollary 2 If the household is composed by one decision maker, household consumption Euler equations simplify to the following standard form:

" 2 2# Ct+1 Qt+1 Ct+1 Ct ln = α0 + α1 ln Rt+1 + α2 ln + α5 ln − ln Ct Qt Cˆ Cˆ " 2 2# Qt+1 Qt +α6 ln − ln + RC C,ˆ Q,ˆ Zˆ + ln (1 + et+1,C ), Qˆ Qˆ " 2 2# Qt+1 Ct+1 Ct+1 Ct ln = δ0 + δ1 ln Rt+1 + δ2 ln + δ5 ln − ln Qt Ct Cˆ Cˆ " 2 2# Qt+1 Qt +δ6 ln − ln + RQ C,ˆ Q,ˆ Zˆ + ln (1 + et+1,Q), Qˆ Qˆ

By means of theorems 3, 4 and corollary 2 it is possible to set up a ﬁrst test to evaluate the ability of the unitary and collective framework to approximate household intertemporal behavior. TEST 1. Under the assumption of rational expectations, Euler equations of households with only one decision maker should not exhibit excess sensitivity to income or other variables known at time t. However, income and other factors known at time t should enter signiﬁcantly household Euler equations, if these variables affect the individual decision power. This test can be implemented estimating separately Euler equations for single and couples after controlling for self selection. The approach of splitting the sample in two groups based on theoretical ground has been ex- tensively used in the literature. For example, Bernanke (1984), Hayashi (1985), Zeldes (1989) and more recently Browning and Chiappori (1998) make use of this strategy. Theorems 3 and 4 make

19 clear that test 1 evaluates the unitary model against the ICM, but it is not possible to distinguish between the full-efficiency and the limited commitment version of the model. However, if test 1 rejects the unitary model, by means of theorems 3 and 4 it is possible to construct an additional test to discern between the two versions of the ICM. TEST 2. Under the assumption of full commitment, the factors Z should not enter the household Euler equations directly, but only as interaction terms with private and public consumption. If limited commitment is a correct specification of household intertemporal behavior, the factors Z should enter both directly and as interaction terms in the household Euler equations. If test 1 rejects the unitary model, test 2 enables one to distinguish between the two versions of the ICM. However, other competing models may be consistent with the outcome of tests 1 and 2. The next corollary to theorems 3 and 4 contains the ingredients to derive a proportionality condition that is unlikely to be fulfilled if the ICM is not a correct characterization of household behavior.

Corollary 3 If the full efﬁciency ICM is a correct representation of household intertemporal behavior, then,

αi,3 δi,3 αi,4 δi,4 = = Ψi,j, = = Φi,j, αj,3 δj,3 αj,4 δj,4 for any pair of factors zi, zj. If the limited commitment ICM is a correct representation of household intertemporal behavior, then,

αi,3 δi,3 αi,4 δi,4 αi,5 δi,5 = = ψi,j, = = φi,j, = = υi,j, αj,3 δj,3 αj,4 δj,4 αj,5 δj,5 for any pair of factors zi, zj.

A test to determine the empirical validity of the full efﬁciency ICM or limited commitment ICM can now be constructed. TEST 3. Euler equations for private consumption and the public good should be estimated simultaneously. Then the proportionality condition can be tested. To perform this test at least two consumption goods and two factors are required.

20 5 Data

In the estimation of household Euler equations two different data sets are employed: the Panel Study of Income Dynamics (PSID) and the Consumer Expenditure Survey (CEX). Both data sets have advantages and disadvantages. The PSID is a true panel, but unfortunately it contains only data on food consumption. The CEX is a survey eliciting detailed information on almost any category of consumption, but it is a rotating panel. In particular, households are followed for only 4 quarters and then dropped from the sample. The next two subsections describe the two data sets.

5.1 PSID

The PSID has gathered data on income and consumption annually each spring from 1968, following the same households and their split-offs over time. In the estimation, I use the data collected from 1975 to 1987. I do not use the data from 1968 to 1974 because the marginal tax rate was not calculated before 1974.6 I do not use data from 1988 to 1993, the last available wave, because the food consumption questions were not asked in 1988 and 1989. Two variables are crucial in the tests derived in the previous section: household total consumption and income. Consumption. The PSID does not contain data on total consumption. The only measure of consumption reported in the survey is annual household expenditure on food. There are advantages and disadvantages in utilizing the PSID. The use of food consumption in the estimation of the Euler equation is justiﬁed only if food is separable from other consumption components.7 It is also well documented the existence of measurement errors in the data contained in the PSID.8 However, the estimation of the Euler equation requires the service ﬂows of consumption goods for a given period. Food expenditure is less likely to contain durable parts. Finally, a legiti- mate question is whether the empirical relevance of the ICM can be determined using only food

6The marginal tax rate can be computed using the NBER’s TAXSIM model. However, this is likely to introduce additional measurement errors. 7There is mixed evidence regarding the separability between food and other consumption goods (see for example Deaton and Muellbauer (1980) and Mankiw, Rotenberg and Summer (1985)) 8See for example Altonji and Siow (1987) and Runkle (1991).

21 consumption. Intuitively any type of consumption should be appropriate. The PSID collects information on three separate components of food expenditure: the cost of food consumed at home; the amount spent in restaurants; the value of food purchased with food stamps. Any empirical work utilizing consumption data from the PSID struggles with the timing of the question concerning food consumption. In fact, it is not clear what period the survey question refers to. The food consumption question in the PSID questionnaire is, ” How much do you spend on the food that you use at home in an average week?” From 1977 this question has been asked following the food stamp question ”Did you receive or buy food stamps, last month?”. If the answer was affirmative, the following question was asked, ”In addition to what you spent on food stamps, did you spend any money on food that you use at home?” and, if yes ”How much?”. Therefore, it seems evident that the surveyors are trying to extract information about food consumption for the period that immediately precedes the interview. Hence, given that the interviews take place in March or April, I interpret the answers as concerning food consumption for the first quarter of the interview year and choose accordingly the timing of the interest rate and of the inflation rates. I also experiment with the alternative interpretation that the consumption question refers to the prior calendar year and obtain similar results9. To have total food consumption, I deflate food consumed at home and away from home by their respective consumer price indexes (CPI) and sum the two real components. Disposable income. Disposable income is computed as total household income, minus taxes, plus transfers. The real after-tax interest rate. The interest rate is the first quarter average of the 1-year Treasury bill rate. I use the marginal tax rate on earned income for husband and wife for year t+1 as estimated by the Survey Research Center of the University of Michigan. The measure of the inflation rate is the growth of overall food CPI between the first quarter of year t and year t + 1.

9Zeldes (1989) and Runkle (1991) give the same interpretation of the question on food consumption, whereas Hall and Mishkin (1982) and Lawrance (1991) use the alternative interpretation.

22 5.2 CEX

Since 1980, the CEX survey has been collecting data on household consumption, income and different types of demographics. The survey is a rotating panel organized by the Bureau of Labor Statistics (BLS). Each quarter about 4500 households, representative of the US population, are interviewed: 80% are reinterviewed the following quarter, while the remaining 20% are replaced by a new randomly selected group. Each household is interviewed at most for four quarters and detailed information are elicited in regard to expenditures for each of the three months preceding the interview, and in regard to income and demographics for the quarter preceding the interview.10 The data used in the estimation cover the period 1980-1995. Consumption. The CEX data set contains monthly data on consumption. Total consumption is computed as the sum of food at home, food out, tobacco, alcohol, other nondurable goods and services such as heating fuel, public and private transportation, personal care and semidurable goods which include clothing and shoes. In particular, from the definition of total consumption I exclude consumer durables, housing, education and health expenditure. Total consumption is deflated using the Consumer Price Indices published by the BLS. Specifically, the price index for the composite good is calculated as a weighted average of individual price indices, with weights equal to the expenditure share for the particular consumption good. Disposable income. Disposable income is computed as total household income, minus taxes. The real after-tax interest rate. The interest rate is the 3-month Treasury bill rate. The marginal tax rate on earned income is computed using the NBER’s TAXSIM model. The measure of the inflation rate is the household specific price index. Rather than employing the short panel dimension of the CEX, I construct synthetic panels. The synthetic panels are constructed using two variables: the year of birth of the head of the household and a dummy equal to 1 if the head is married and 0 otherwise. All households are assigned to one of these cells. I then average the variables of interest over all the households belonging to a given cohort observed in a given quarter. To avoid unnecessary overlapping between quarters, for each

10Each household is interviewed for 5 quarter, but the ﬁrst interview is used to make contact and no information is publicly available.

23 Table 1: cohort deﬁnition for couples and singles cohort Year of Birth Age in 1980 Average Cell Size Average Cell Size Used in Couples Singles Estimation

1 1905-1909 71-77 61 71 no 2 1910-1914 66-72 118 107 no 3 1915-1919 61-67 153 113 yes 4 1920-1924 56-62 178 119 yes 5 1925-1929 51-57 193 118 yes 6 1930-1934 46-52 196 102 yes 7 1935-1939 41-47 210 103 yes 8 1940-1944 36-42 248 128 yes 9 1945-1949 31-37 312 170 yes 10 1950-1954 26-32 322 205 yes 11 1955-1959 21-27 319 238 yes 12 1960-1964 16-20 263 241 yes 13 1965-1969 11-15 193 205 no household in each quarter, I use only the consumption data for the month preceding the interview and drop the data for the previous two months. Table 5.2 contains a description of the cohort.

5.3 Selection of the PSID Sample

The PSID data set contains a subsample in which low-income households are overrepresented. I eliminate this group to have a representative sample of the US economy. I include split-off families as separate families, but eliminate from the sample the observations associated to their ﬁrst year as a new family to reduce the measurement errors. I eliminate outliers from the sample.

Speciﬁcally, an observation is considered an outlier if ln (ci,t+1/ cit) > 1.1, i.e. if the level of food consumption rose or fell by more than a factor of 3 in a year. I only include in the sample

24 households whose head is between the ages of 19 and 75. Finally, I do not use an observation if consumption or income data were estimated by the Survey Research Center.

5.4 Selection of the CEX Sample

I exclude from the sample rural households, households living in student housing, household in which the head is younger than 19 and older than 75 and households with incomplete income responses. I drop households experiencing a change in marital status. Finally a cohort is dropped if the cell size is lower than 100.

6 Econometric Issues

The household Euler equations derived in section 4 contain the logarithm of the expectation error.

By assumption, the expectation error term ei,t+1 has mean zero and is orthogonal to the information set at time t. However, it could be correlated across households because of macro shocks. To solve this problem, I will assume, as in Zeldes (1989), that (1 + ei,t+1) can be decomposed in the product of two orthogonal components with mean zero: an aggregate component that is common a to all households, 1 + et+1 , and an idiosyncratic component, (1 +e ¯i,t+1). By construction, ln (1 +e ¯i,t+1) does not necessarily have zero expectation, implying that the error term of the log- linearized Euler equations is not, in general, a mean zero variable. To solve this problem, notice that the expected value of the second order Taylor expansion of ln (1 +e ¯it+1) is given by,

1 E [ln (1 +e ¯ )] = − σ2 , t i,t+1 2 i,t+1

2 1 2 where σi,t+1 = V art [¯ei,t+1]. As a consequence ηi,t+1 = ln (1 +e ¯i,t+1) − 2 σi,t+1 has mean zero. Taking into account all this and assuming that V art [¯ei,t+1] is constant over time and across families, the log-linearized Euler equations can be estimated consistently including V art [¯ei,t+1] in the constant. The likely presence of aggregate shocks implies that Euler equations can be consistently estimated only if households are observed over a long period of time. Indeed, if this condition is satisﬁed, aggregate expectation errors are averaged out. In the PSID, most households

25 are observed for the entire sample period. Therefore aggregate errors should be averaged out.11 In regard to the CEX, since I construct synthetic cohorts and these cohorts are followed for the whole sample period, the aggregate error should not affect the estimation results. This solves the small T problem discussed by Chamberlain (1984). In the theoretical part of the paper, a public good is always included in the Euler equations for two reasons. Some of the tests derived in section 4 requires the joint estimation of Euler equations for two different goods. More important, there is at least one public good that is likely to affect household intertemporal behavior: consumption of children. In most households, children are not decision makers. Consequently, they do not enter the household decision process directly. However, their consumption should be in the utility functions of their parents. Unfortunately, the consumption of children is not measured in the PSID and CEX. As a partial solution, I use the number of children as a proxy for consumption of children. In the estimation of Euler equations with the CEX, two demographic variables are always included: the age of the head of the household and family size growth. When the PSID data set is used, family size growth is replaced by annual food needs growth, since only food consumption is observed. I will assume that the number of children, age and family size at time t + 1 are known to the household at time t, implying that they will be uncorrelated with the error term. In general the after-tax interest rate will not be known at time t and will probably be correlated with the expectation error because of the correlation between consumption and the marginal tax rate at t + 1. To solve this problem, I instrument the after-tax interest rate. Since total income may be affected by measurement errors, this variable is also instrumented. Under the assumption of rational expectations, variables known at time t should be valid instruments. Measurement errors may introduce dependence between variables known at time t and future variables, even under rational expectations. To avoid this problem I only use variables known at t − 1. The following instruments are used in the estimations: the second, third and fourth lag of the nominal interest rate; the second, third and fourth lag of consumption growth; the second, third and fourth lag of

11The PSID sample is not a balanced panel, implying that different households are in the survey for different periods.

26 family size; the second, third and fourth lag of children.12 In the derivation of household Euler equations, labor supply decisions are never considered. The implicit assumption is that consumption and leisure are separable. The PSID contains only food consumption. Consequently, I will assume separability between food and other goods. When Euler equations are estimated using the CEX data set, all consumption goods are aggregated in a composite good. Hence, I will assume that all the conditions of the composite good theorem are satisﬁed. To perform the tests, I divide the sample in two groups using marital status at time t. The ﬁrst group is composed by singles, widows, divorcees and separated couples. The second group contains married and cohabiting couples. If the Euler equations are estimated using OLS, the estimates are likely to be affected by self selection bias. Therefore, I employ a Heckman two-step estimator adding the inverse Mill’s ratio to the Euler equations derived in this section. The main implication of the theoretical part is that the family structure is extremely important for the determination of the Euler equation, as it affects the household decision process. While it is beyond the scope of this paper to model theoretically how households choose between different family structures, from an econometric point of view it is crucial to take it into consideration.13 m Let Dj,i,t be a dummy variable equal to 1 if agent j is married or cohabiting in household i. h s i h m i Let E Vj,i(X, t) |Ft and E Vj,i (X, t) |Ft be, respectively, the expected value of being single s and married today for agent j in household i, where X is the set of variables affecting Vj,i and m Vj,i and Ft is the information known at time t. I will make the simplifying assumption that h s i h m i E Vj,i(X, t) |Ft and E Vj,i (X, t) |Ft are linear functions of current variables, Xt.

s s s Vj,i(X, t) = Xj,i,tβt + ej,i,t

m m m Vj,i (X, t) = Xj,i,tβt + ej,i,t h i h ˆ m with E ej,i,t |Ft = 0, h = s, m. Denote with Vj,i (X, t) the best feasible match available to

12In the estimations with the quarterly CEX data, three seasonal dummies are also included. 13For a discussion on the variables affecting the marriage decision see for instance Lillard et al. (1995) and Rindfuss and VandenHeuvel (1990).

27 agent j, i.e. ˆ m ˆm m m Vj,i (X, t) = Xj,i,tβt +e ˆj,i,t = max Vj,k(X, t) k∈K where K is the set of feasible households available to agent j. Then the selection decision can be written as follows,

m ˆm m s s Dj,i,t = 1 if Xj,i,tβt +e ˆj,i,t − Xj,i,tβt + ej,i,t ≥ 0 (14)

m Dj,i,t = 0 otherwise.

m Given the deﬁnition of Dj,i,t, the Euler equation for household i can be written as follows:

s s s s GCi,t+1 = δ + Yi,tγ + ηi,t+1 if ej,i,t < −Xj,i,tβt (15)

m m m m GCi,t+1 = δ + Yi,tγ + ηi,t+1 otherwise (16)

m s ˆm s where eit =e ˆj,i,t − ej,i,t, βt = βt − βt and Yi,t is the set of variables characterizing the Euler h equation. Under the assumption that for each t and for h = s, m,(ηi , ei) is a bivariate normal distribution with mean vector 0 and covariance matrix,   2 ση,h ρη,h   1 the household Euler equations can be consistently estimated applying a standard argument to equations (15) and (16) to derive,14

s s s s s GCi,t+1 = δ + Yi,tγ + ρη,sση,sλi,t + i,t+1 (17)

m m m m m GCi,t+1 = δ + Yi,tγ + ρη,mση,mλi,t + i,t+1. (18)

s h s m i m h m m i h where ρη,sση,sλi,t = E ηi,t+1 |Dit = 0,Yi,t , ρη,mση,mλi,t = E ηi,t+1 |Dit = 1,Yi,t , λi,t is h the inverse Mill’s ratio and i,t+1 are the new residuals, with zero conditional mean. The two-stage estimation procedure can now be spelled out. First, estimates of βt are obtained by estimating T

14For a thorough discussion of the Heckman two-steps estimator see Heckman (1976, 1979), Lee (1978).

28 cross-sectional probit speciﬁcations by maximum likelihood. 15 I will then estimate equations ˆs ˆm s m (17) and (18) by OLS, substituting the estimated λi,t and λi,t for λi,t and λi,t. To calculate the asymptotic covariance matrix for the parameter estimates, I follow Chamber- h 2 lain (1984) and MaCurdy (1982). Let ψ = ρηhσηh. Moreover, let θ, Λ, and Z be deﬁned as follows:

θ = (β1, ..., βT ) ξk = γk, ψh , k = s, m

Z = Y, λˆh " # ∂λˆk ∂λˆk ∂λˆk Λk = 1 , 2 , ..., NT , ∂θ ∂θ ∂θ where NT is the total number of observations. Then the asymptotic covariance matrix for the parameter estimates of equations (17) and (18) is,

ˆk 0 −1 0 h k 2 2 k ˆ k0i 0 −1 V ar ξ = Z Z Z V ar η + σηkρηkΛ V ar θ Λ Z Z Z where V ar ηk is the covariance matrix of the residual of the standard Euler equations. Since the first step estimates are obtained using T cross-sectional probits, I follow Chamberlain (1984) and MaCurdy (1982) to determine V ar θˆ . Specifically, let Q (θ) be the likelihood function of the probit. Let ψ, ∆ and J be defined as follows: ∂Q ψ (X, θ) = ∂θ

∆ = E ψ (X, θ) ψ (X, θ)0

∂ψ (X, θ) J = E . ∂θ0

15The estimation of T cross-sectional probit specifications by maximum likelihood enables one to allow for a general structure of the variance-covariance matrix without computing multiple integrals. Since the inverse Mill’s ratio is computed using a panel, it is important to allow for this generalization. The cost is a loss in efficiency. When the CEX data are used in the estimation, the first step estimates are computed by means of a single probit, because I do not have enough observations each period to compute T cross-sectional probits.

29 Then16, V ar θˆ = J −1∆J −1. (19)

Finally, to allow each family to have a different and unrestricted covariance structure, I follow White (1984) and Keane and Runkle (1992) and estimate V ar ηk as follows,

N k 1 X 0 Vd ar η = IN ⊗ uiu , (20) N i i=1 where ui is the vector of estimated Euler equation residuals for family i. Depending on the data set employed in the estimation, two different exclusion restrictions are used. With the PSID, it is assumed that the number of siblings of the head of the household and race dummies are correlated with the marriage decision, but not with the consumption decision. The CEX does not report the number of siblings of the head. Dummies indicating the origin or ancestry of the head are used to proxy for the number of siblings and the race dummies.

7 Results

Tables 4 and 5 in the appendix contain a summary of means for the main variables. Tables 2 and 3 contain the estimates for singles and couples when the log of household total income is included in the Euler equations as a proxy for the factors affecting individual decision power.

7.1 Test 1

Group 1. The ICM has one clear prediction: the household Euler equation should not be violated for families with one adult. Tables 2 and 3 suggest that this hypothesis is not rejected using both the PSID and the CEX. The coefficient on the log of available income at time t is not significant and small in magnitude. The coefficient on the interest rate is positive. The coefficient on age is negative. The coefficient on family size growth is positive as expected and strongly significant.

16Since θˆ is estimated by means of T cross-sectional probits, J will be a block diagonal matrix with the generic t-block equivalent to the information matrix at t.

30 The coefficient on the public good, number of children, is negative. The self selection correction term is not significant. Group 2. Euler equations for couples exhibit excess sensitivity to income using the PSID as well as the CEX. Specifically, the coefficient on the log of disposable income is significant and, more important, its magnitude is two and one half times the magnitude of the coefficient obtained for group 1 in the CEX, and almost ten times larger in the PSID. The coefficient on age is negative. The coefficient on family growth is positive and strongly significant. Finally the coefficient on the interest rate is positive as expected. As for groups 1 the sample selection coefficient is not significant.

7.2 Test 2

To be added.

7.3 Test 3

To be added.

31 Table 2: PSID data set. Group 1: singles, divorcees, separated and widows. Group 2: married and cohabiting couples (standard errors in brackets).

Euler Equation Estimates for Two Subgroup Independent Variable Group 1 Group 2 age of head -0.001 -0.0011 [0.00023] [0.00043] growth in annual food needs 0.39 0.27 [0.064] [0.056] growth in n. of children -0.0026 0.0008 [0.0049] [0.0021] real after-tax interest rate 0.53 0.48 [0.31] [0.14] log of disposable income -0.004 -0.039 [0.021] [0.014] inverse Mill’s ration 0.0088 -0.012 [0.012] [0.028] number of observations 3727 15258 number of families 797 2178

32 Table 3: CEX data set. Group 1: cohorts composed by singles, divorcees, separated and widows. Group 2: cohorts composed by married and cohabiting couples (standard errors in brackets).

Euler Equation Estimates for Two Subgroup Independent Variable Group 1 Group 2 age of head -0.0003 -5.5e-06 [0.0003] [0.0002] growth family size 0.28 0.84 [0.13] [0.27] growth in n. of children -0.021 -0.041 [0.026] [0.018] real after-tax interest rate 0.63 0.87 [0.19] [0.20] log of disposable income 0.015 0.037 [0.015] [0.013] inverse Mill’s ration -0.013 0.003 [0.018] [0.015] number of observations 464 544 number of cohorts 10 10

33 8 Conclusions

In the paper, each household is represented as a group of agents making joint decisions. By means of this framework, it is shown that, under the assumption of full efficiency, the household can be represented by means of a unique utility function if and only if all members have identical discount factors and HARA preferences with identical shape parameter. If these conditions are not satisfied, the intertemporal allocation of resources will depend on the distribution of decision power. In particular, the traditional Euler equation will not be satisfied. For the limited commitment case, it is shown that, even if these restrictions are fulfilled, the household Euler equation is replaced by an inequality. It is also shown that the full efficiency ICM has different predictions for Euler equations from the limited commitment ICM: under full commitment, the factors affecting individual decision power should enter household Euler equations only as interaction terms with household consumption; if the assumption of full commitment is not satisfied, those factors should enter household Euler equations both directly and as interaction terms with household consumption. The standard unitary framework as well as the ICM predict that the traditional Euler equation should not be violated for households with one decision maker. The two frameworks have opposite implications for households with several decision makers, however. The unitary model predicts that the standard Euler equation should not be violated, whereas the ICM predicts that it should be rejected. These predictions are tested using the PSID as well as the CEX. I find that Euler equations are not violated for households with one adult. However, Euler equations are strongly rejected for couples with both members in the labor force. The theoretical and empirical results indicate that it is crucial to model households with several decision makers as groups of agents with heterogeneous preferences and different decision power.

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39 A Appendix

A.1 Proof of theorem 1

(Sufficiency) Rubinstein (1974). (Necessity) The second part of Theorem 1 will be proved in four steps. First, the intertemporal decision process can be divided in two steps. In the first step, for an arbitrary sequence of public consumption, {Qt,ω}, the household decides the optimal sequence of ci , s i=1,...,n private consumption and savings, t,ω t,ω ω∈Ω,t∈T . In the second step, given the optimal sequence of private consumption and savings for any given sequence of the public good, the household chooses the optimal sequence of the public good. The remaining parts of the theorem deal with the first step of the intertemporal decision process. Specifically, all the results should be interpreted conditional on the public good. To simplify the notation I will suppress the dependence on the public good.

i∗ i∗ ω∈Ω,t∈T Lemma 1 Let c (t, ω) , s (t, ω) i=1,...n be the optimal household allocation. Assume i0 i {µi} 0 and limu (x) = ∞. Moreover suppose that u is continuous, strictly increas- i=1,...,n x→0 ing, strictly concave and twice continuously differentiable. Then there exist prices {p (t, ω)}ω∈Ω,t∈T i i i∗ i∗ ω∈Ω,t∈T and transfers W i=1,...n such that {p (t, ω)}ω∈Ω,t∈T , W i=1,...n and c (t, ω) , s (t, ω) i=1,...n is an Arrow-Debreu equilibrium with transfers, or equivalently, i∗ i∗ ω∈Ω,t∈T (i) for each i = 1, ...n, c (t, ω) , s (t, ω) i=1,...n solves

T X i t i max E βi ui c (t, ω) {ci∗,si∗} ω∈Ω,t∈T t=0 T X Z s.t. p (t, ω) ci (t, ω) dω = W i. t=0 Ω (ii) for each t, ω,

n X ci∗ (t, ω) = Y (t, ω) + R (t) s (t − 1, ω) − s (t, ω); i=1 (ii) for each s, t, ω,

40 p (t, ω) R (t) s∗ (t − 1, ω) − p (t − 1, ω) s∗ (t − 1, ω) ≥

p (t, ω) R (t) s (t − 1, ω) − p (t − 1, ω) s (t − 1, ω) .

Proof. The technology y = f (x) = Rx is convex. The utility functions are strictly concave and increasing. Let Xi = ci ∈

Lemma 2 Household exact aggregation is satisﬁed if and only if for each pair (t, ω) individual Engel curves are linear with identical slope, i.e.

ci t, ω; W i = ai (t, ω) + b (t, ω) W i ∀i, t, ω.

Proof. Lemma 1 implies that the household problem can be written as a static problem. By Gorman (1951), in a static framework, exact aggregation is satisﬁed if and only if for each consumption good individual Engel curves are linear with identical slope. In this framework this is equivalent to ci t, ω; W i = ai (t, ω) + b (t, ω) W i ∀i, t, ω.

The next lemma establishes conditions on preferences for the Engel curves to be linear with identical slope and hence for exact aggregation to apply.

41 Lemma 3 Individual Engel curves are linear with identical slope if and only if household members have identical discount factors, individual utility functions are HARA with identical curvature parameter.

Proof. By lemma 1, the solution of the household problem can be determined solving for i = 1, ..., n the following program:

T Z X t i max βi ui c (t, ω) Fi (dω) (21) {ci∗,si∗} ω∈Ω,t∈T t=0 Ω T X Z s.t. p (t, ω) ci (t, ω) dω = W i. t=0 Ω

By assumption, for each i = 1, ..., n, the probability measure Fi (ω) has a density fi (ω). There- fore the program (21) can be written in the form,

T Z X t i max βi ui c (t, ω) fi (ω) dω {ci∗,si∗} ω∈Ω,t∈T t=0 Ω T X Z s.t. p (t, ω) ci (t, ω) dω = W i. t=0 Ω The corresponding Lagrangian is,

T Z X t i i L = βi ui c (t, ω) fi (ω) − λip (t, ω) c (t, ω) dω, t=0 Ω where λi is the Lagrangian multiplier. By the calculus of variations, the following equations are i∗ necessary conditions for c (t, ω) ω∈Ω,t∈T to be a solution of the household problem:

t 0 i βi fi (ω) ui c (t, ω) = λip (t, ω) ∀t, ω.

Hence, 0 i fˆi (t, ω) u c (t, ω) p (t, ω) i = ∀t, k, ω, ω0, ˆ 0 0 i 0 p (t + k, ω0) fi (t + k, ω ) ui (c (t + k, ω )) ˆ t where fi (t, ω) = βi fi (ω). It is now possible to apply theorems 2 and 3 in Brennan and Kraus (1978) to conclude that individual Engel curves are parallel straight lines if and only if the following to conditions are jointly satisﬁed:

42 (i) all household members have HARA utility functions with identical curvature parameter;

(ii) fˆi (t, ω) = fˆj (t, ω) for any i, j, t, ω. ˆ t Since fi (t, ω) = βi fi (ω), condition (ii) is satisﬁed if and only if all agents have identical discount factors and identical probability measure over Ω.

A.2 Proof of corollary 1

In order to prove corollary 1 and theorems 2 the following results are required. The following lemma is theorem 198 in Hardy, Littlewood and Polya (1934).

Lemma 4 Let x1 and x2 be nonnegative random variables deﬁned on (Ω, F) and ﬁnite almost 1 R γ γ everywhere. Set x = x1 + x2. If γ ∈ R and γ < 0, then the function x dP is concave in x or equivalently (given homogeneity of degree 1),

1 1 1 Z γ Z γ Z γ γ γ γ x dP ≥ x1 dP + x2 dP .

Let A and B be two constants. If P {ω ∈ Ω: Ax1 (ω) = Bx2 (ω)} < 1, and

P {ω ∈ Ω: x1 (ω) = x2 (ω) = 0} = 0, then

1 1 1 Z γ Z γ Z γ γ γ γ x dP > x1 dP + x2 dP .

Proof. See Hardy, Littlewood and Polya (1934). 0 0 Following Ash (1972), let a random object Xt be a mapping Xt : (Ω, F) → (Ω , F ), for 0 0 some measurable spaces (Ω, F) and (Ω , F ). Define σ hXti to be the σ-field generated by Xt, i.e. 0 the minimum σ-field that makes Xt measurable between F and F . The following lemma gives a characterization of the filtration {Ft}t∈T in terms of random variables.

Lemma 5 Let {Ft}t∈T be the ﬁltration associated with F. Then for every t ∈ T there exists a 0 0 random object Xt : (Ω, F) → (Ω , F ) such that σ hXti = Ft.

Proof. Deﬁne Xt as follows,

43 (i) Xt : (Ω, F) → (Ω, Ft);

(ii) Xt is the identity map, i.e. Xt (ω) = ω ∀ω ∈ Ω.

−1 −1 ∀A ∈ Ft, Xt (A) = {ω ∈ Ω: Xt (ω) ∈ A} = A by construction, which implies Xt (Ft) =

Ft. −1 Then by theorem 6.4.2 in Ash (1972), σ hXti = Xt (Ft) = Ft. The following lemma claims that a conditional expectation given a σ-ﬁeld can be written as a function of a well-chosen random object.

Lemma 6 Let h (ω) = E [Y |Ft ] and g (x) = E [Y |Xt = x]. Then there exists a random object

Xt : (Ω, F) → (Ω, Ft) such that,

h (ω) = (g ◦ Xt)(ω) ∀ω ∈ Ω.

Proof. By Lemma 5, there exists a random object Xt : (Ω, F) → (Ω, Ft) such that σ hXti = −1 Xt (Ft) = Ft.

Let g (x) = E [Y |Xt = x]. Then by theorem 6.4.3 in Ash (1972),

g (Xt (ω)) = E [Y |σ hXti] = E [Y |Ft ] = h (ω) ∀ω ∈ Ω.

Let Xt be the random object deﬁned in lemma 5 and x a realization of Xt.

Lemma 7 Assume Y : (Ω, F) → (R, R) is a nonnegative random variable. For every x and B ∈ R, let P (x, B) be a probability measure in B for each ﬁxed x and a Borel measurable function of x for each ﬁxed B, i.e. P (x, B) is the conditional distribution of Y given Xt = x. Let γ ∈ R. Then Z γ γ E [Y |Xt = x] = Y P (x, dy) .

R Proof. The argument of the proof follows closely Ash (1972) section 6.3.5 part (d).

By construction, Y : (Ω, F) → (R, R) and Xt : (Ω, F) → (Ω, Ft) .

44 0 0 Let Ω = R × , F = R × F, π1 (x, y) = x and π2 (x, y) = y.

∀A ∈ Ft let PXt [A] = P [{ω ∈ Ω: Xt (ω) ∈ A}], i.e. PXt is the probability on (Ω, Ft) induced from P by Xt.

Deﬁne Pxy as follows, Z

Pxy [A, B] = P (x, B) dPXt (x) . A

0 By the product measure theorem, Pxy is the unique probability measure on F determined by

PXt and P (x, ·). γ γ R γ Consider (π2) = y :( , R) → ( , R). y dP (x, dy) exists by Y (ω) ≥ 0 ∀ω ∈ Ω. R R R Then, by Fubini’s theorem and deﬁnition of Pxy, Z Z γ γ π2 dPxy = π2 I{Xt∈A}dPxy = 0 {Xt∈A} Ω Z Z γ = π2 (x, y) IA (x) P (x, dy) dPXt (x) =

Ω R   Z Z γ =  y P (x, dy) dPXt (x) .

A R By deﬁnition of conditional expectation, this implies, Z γ γ γ E [π2 |Xt = x] = E [Y |Xt = x] = Y P (x, dy) .

Finally, for (4) to be well-defined we have to assign a conditional probability governing the transition between t and t + 1. Let Xt be the random object defined in lemma 5 and x a particular realization of Xt. For every x and B ∈ F, let P (x, B) be a probability measure in B for each fixed x, and a Borel measurable function of x for each fixed B. Consider a random −1 variable Y : (Ω, F) → (R, R). For every x and A ∈ R, let PY (x, A) = PY x, Y (A) =

PY (x, {ω ∈ Ω: Y (ω) ∈ A}), i.e. PY (x, A) is the conditional distribution induced from P of a random variable Y : (Ω, F) → (R, R), given Xt = x.

45 It is now possible to prove corollary 1. Proof. I will prove the theorems for n = 2. The proof for an arbitrary n can be obtained iterating n − 1 times the same argument. Given the assumption of ISHARA households, without loss of generality I will assume βRt+1 = 1. By proposition 1 the solution of the ICM (2) is equivalent to the solution of the two-stage budgeting model. The assumption of HARA preferences ensures that second stage (4) has an interior solution. The ﬁrst order conditions for the second stage (4) imply, i i0−1 i0 i ct = u E u ct+1 |Ft for i = 1, 2.

Given assumption 2,

 − 1  " i −γ #! γ −1 c ui0 E ui0 ci |F = γ E a + t+1 |F − a . t+1 t  i γ t i

ci Fix an ω ∈ Ω. Deﬁne zi = a + t+1 . By Lemma 7, t+1 i γ

i h i −γ i h i −γ i h (ω) = E zt+1 |Ft (ω) = E zt+1 |Xt (ω) = x .

i i ct+1 By deﬁnition of permissible income process, z = a + ≥ 0 ∀ω, i = 1, 2. Let P i (x, B) = t+1 i γ z x Pi (B). For any B ∈ R,

x x i x i Pi (B) = P z ∈ B = P ω ∈ Ω: z (ω) ∈ B ,

x x i i.e. Pi is a probability measure induced from P by z . By Lemma 7 and the change of variable theorem, Z Z h i −γ i i −γ x i −γ x E zt+1 |Xt (ω) = x = zt+1 dPi = zt+1 (ω) dP (ω) . (A.1)

R The ﬁrst order conditions of the ﬁrst stage (5) imply,

1 2 P ω ∈ Ω: Azt (ω) = Bzt (ω) = 1 ∀ (t, ω) , (A.2)

46 for some constant A and B. This condition is simply the Borsh rule. Lemma 4, equations (A.1) and (A.2) imply,

 1   1   - γ  - γ Z Z 1 2  1 -γ x   2 -γ x  c + c = γ  z (ω) dP (ω) -a1 + γ  z (ω) dP (ω) -a2 = t t  t+1    t+1  

 1 1   − γ  − γ Z Z  1 −γ x 2 −γ x  = γ  z dP  +  z dP  − (a1 + a2) =  t+1   t+1  

 1   − γ Z  1 2 −γ x  = γ  z + z dP  − (a1 + a2) =  t+1 t+1  

 1   − γ Z 1 2 −γ  ct+1 (ω) + ct+1 (ω) x  = γ  a1 + a2 + dP  − (a1 + a2) =  γ  

0−1 0 1 2 = U E U ct+1 + ct+1 |Ft .

Applying U 0 to both sides, 0 0 U (ct) = E U (ct+1) |Ft .

A.3 Proof of theorem 2

Proof. The FOC’s for the limited commitment ICM imply the following:

" ˆ1 ! ˆ1 ˆ2 # 0 k 0 k ∂u k ∂u u (c1) = E 1 + t+1,ω u (c1 )+ t+1,ω 1,t+1,ω + t+1,ω 2,t+1,ω (22) 1 t t ˆ 1 1 t+1 ˆ 1 ˆ 1 Kt Kt ∂st Kt ∂st " ˆ2 ! ˆ1 ˆ2 # 0 k 0 k ∂u k ∂u u (c2) = E 1 + t+1,ω u (c2 )+ t+1,ω 1,t+1,ω + t+1,ω 2,t+1,ω , 2 t t ˆ 2 2 t+1 ˆ 2 2 Kt Kt ∂st Kt ∂st

ˆ1 where kt+1,ω ≥ 0 is the Kuhn-Tucker multiplier associated with the participation constraint ad- ˆ i Pt ˆi justed for the discount factor and the probability function and K (t, ω) = τ=0 k (τ, ω) ≥ 0.

47 ∂u Assume i,t+1,ω = 0 for every i. Then (22) simpliﬁes to, ∂st (ω) " ! # kˆ1 u10 c1 = E 1 + t+1,ω u10 c1 |F t ˆ 1 t+1 t Kt " ! # kˆ2 u20 c2 = E 1 + t+1,ω u20 c2 |F , t ˆ 2 t+1 t Kt which implies,

1 10−1 10 1 ct ≤ u E u ct+1 |Ft 2 20−1 20 2 ct ≤ u E u ct+1 |Ft

Given the assumption of HARA preferences,

 − 1  " i −γ #! γ −1 c ui0 E ui0 ci |F = γ E a + t+1 |F − a . t+1 t  i γ t i

ci Fix an ω ∈ Ω. Deﬁne zi = a + t+1 . The ﬁrst order conditions of (3) imply, t+1 i γ

P [{ω ∈ Ω: Az1 (ω) = Bz2 (ω)}] < 1. (A.2’)

Lemma 4, equations (A.1) and (A.2’), and the assumption on preferences imply,  1   1   - γ  - γ Z Z 1 2  1 -γ x   2 -γ x  c + c ≤ γ  z (ω) dP (ω) -a1 + γ  z (ω) dP (ω) -a2 t t  t+1    t+1  

 1 1   − γ  − γ Z Z  1 −γ x 2 −γ x  = γ  z dP  +  z dP  − (a1 + a2)  t+1   t+1  

 1   − γ Z  1 2 −γ x  < γ  z + z dP  − (a1 + a2)  t+1 t+1  

 1   − γ Z 1 2 −γ  ct+1 (ω) + ct+1 (ω) x  = γ  a1 + a2 + dP  − (a1 + a2)  γ  

48 0−1 0 1 2 = U E U ct+1 + ct+1 |Ft .

Applying U 0 to both sides and using the fact that U 0 is decreasing,

0 0 U (ct) > E U (ct+1) |Ft . ∂u Pn ˆi i,t+1,ω Assume i=1 kt+1,ω ≥ 0. Then by (22), ∂st (ω) " ˆ1 ! # 0 k 0 h 0 i u (c1) ≥ E 1 + t+1,ω u (c1 ) ≥ E u (c1 ) 1 t t ˆ 1 1 t+1 t 1 t+1 Kt " ˆ2 ! # 0 k 0 h 0 i u (c2) ≥ E 1 + t+1,ω u (c2 ) ≥ E u (c2 ) , 2 t t ˆ 2 2 t+1 t 2 t+1 Kt which implies,

1 10−1 10 1 ct ≤ u E u ct+1 |Ft 2 20−1 20 2 ct ≤ u E u ct+1 |Ft . ∂u Hence, applying the same argument as for i,t+1,ω = 0 for every i, it follows, ∂st (ω) 0 0 U (ct) > E U (ct+1) |Ft .

A.4 Proof of theorem 3 and 4

Proof. The one-variable functions ϑ1 : I1 → R and ϑ2 : I2 → R are deﬁned as follows:

ϑ1(t) = φ1(tC,ˆ tQ,ˆ tZˆ)

ϑ2(t) = φ2(tC,ˆ tQ,ˆ tZˆ) where I1 = (−a, a) and I2 = (−b, b). Applying the one-variable Taylor’s formula with remainder,

0 00 2 ϑi(t) = ϑi(0) + ϑi(0)t + ϑi (0)t + ri(t), for i = 1, 2, (23) with Z t 1 3 000 ri(t) = (t − s) ϑi (s)ds. 3! 0

49 Applying the chain rule, we have, ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ∂φi(tC, tQ, tZ) ∂φi(tC, tQ, tZ) X ∂φi(tC, tQ, tZ) ϑ0 (t) = Cˆ + Qˆ + zˆ i ˆ ˆ ∂zˆ i ∂C ∂Q j i 2 ˆ ˆ ˆ 2 ˆ ˆ ˆ 2 ˆ ˆ ˆ 00 ∂ φi(tC, tQ, tZ) 2 ∂ φi(tC, tQ, tZ) 2 X ∂ φi(tC, tQ, tZ) 2 ϑ (t) = Cˆ + Qˆ + zî i ˆ2 ˆ2 ∂zˆ 2 ∂C ∂Q j i 2 ˆ ˆ ˆ 2 ˆ ˆ ˆ 2 ˆ ˆ ˆ X ∂ φi(tC, tQ, tZ) X ∂ φi(tC, tQ, tZ) X ∂ φi(tC, tQ, tZ) + QˆCˆ + + zˆ Cˆ + zˆ Qˆ ˆ ˆ ˆ i ˆ i j ∂Q∂C j ∂zî∂C j ∂zî∂Q

Hence, from (23), with t = 1,

2 2 ∂φi(0) ∂φi(0) X ∂φi(0) ∂ φi(0) 2 ∂ φi(0) 2 φi(C,ˆ Q,ˆ Zˆ) = φi(0) + Cˆ + Qˆ + zˆi + Cˆ + Qˆ ˆ ˆ ∂zˆ ˆ2 ˆ2 ∂C ∂Q j i ∂C ∂Q (24) 2 2 2 2 X ∂ φi(0) ˆ2 X ∂ φi(0) X ∂ φi(0) i X ∂ φi(0) i + z + CˆQˆ + Cˆzˆ + Qˆzˆ + Ri(C,ˆ Q,ˆ Zˆ) ˆ2 i ∂C∂ˆ Qˆ ∂C∂ˆ zˆ ∂Q∂ˆ zˆ j ∂zi j j i j i

Finally by deﬁnition of φi(C,ˆ Q,ˆ Zˆ), we have,

∂φ V ∂κ ∂2φ V V − V V ∂κ 1 = Cκ z , 1 = C CCκ Cκ CC Cz , (25) i ˆ 2 i ∂zî VC ∂zi ∂C∂zî VC ∂zi ∂2φ V V − V V ∂κ ∂φ V ∂κ 1 = C CQκ Cκ CQ Qz , 2 = Qκ z , (26) ˆ 2 i i ∂Q∂zî VC ∂zi ∂zî VQ ∂zi ∂2φ V V − V V ∂κ ∂2φ V V − V V ∂κ 2 = C QCκ Qκ QC Cz , 2 = Q QQκ Qκ QQ Qz , (27) ˆ 2 i ˆ 2 i ∂C∂zî VQ ∂zi ∂Q∂zî VQ ∂zi where κ = µ if the full efficiency ICM is considered and κ = M if the limited commitment ICM is considered. Under the assumption of rational expectations, the household Euler equations can be written in the form,

VC (Ct+1,Qt+1, κ(Z)) βRt+1 = 1 + et+1 VC (Ct,Qt, κ(Z)) where et+1 is the expectation error. Taking logs and using φ1 = ln VC and φ2 = ln VQ, we have,

φi(Cˆt+1, Qˆt+1, Zˆt+1) − φi(Cˆt, Qˆt, Zˆt) = − ln β − ln Rt+1 + ln(1 + et+1) for i = 1, 2.

50 Table 4: PSID summary statistics Variable Mean for Singles Mean for Couples

consumption growth -0.014 -0.0005 age of head 42.6 41.8 growth in annual food needs -0.0029 -0.0039 annual real consumption 2999.2 4832.7 annual after tax income 16806 30237 number of observations 3205 15366 number of families 716 2115

Consider ﬁrst the full efﬁciency ICM. The relative Pareto weight µ is constant over time. Moreover the vector of factors Z is also constant over time. Hence, from (24), (25), (26), (27), ˆ C ˆ Q C = ln E[C] and Q = ln E[Q] , the result follows. Consider the limited commitment ICM. Note that,

Mt+1 = Mt + µt+1 for any (t, ω). (28)

Consequently, ∂M ∂M ∂µ t+1 = t + t+1 for any (t, ω). (29) ∂zi ∂zi ∂zi ˆ C ˆ Q Hence, from (24), (25), (26), (27), C = ln E[C] and Q = ln E[Q] , the result follows.

A.5 Summary statistics

Tables 4 and 5 contains sample means of the main variables used in the estimation.

51 Table 5: CEX summary statistics Variable Mean for Singles Mean for Couples consumption growth 0.0004 -0.0015 age of head 46.6 46.7 growth in family size -0.0009 -0.0031 monthly real consumption 552.1 882.1 annual after tax income 15959 29063 number of observations 908 906